Rigidity of Graphs and Frameworks Bill Jackson School of - - PowerPoint PPT Presentation

Rigidity of Graphs and Frameworks

Bill Jackson School of Mathematical Sciences Queen Mary, University of London England DIMACS, 26-29 July, 2016

Bill Jackson Rigidity of Graphs and Frameworks

Bar-and-Joint Frameworks

A d-dimensional bar-and-joint framework is a pair (G, p), where G = (V , E) is a graph and p is a map from V to Rd.

Bill Jackson Rigidity of Graphs and Frameworks

Bar-and-Joint Frameworks

A d-dimensional bar-and-joint framework is a pair (G, p), where G = (V , E) is a graph and p is a map from V to Rd. We consider the framework to be a straight line realization of G in Rd in which the length of an edge uv ∈ E is given by the Euclidean distance p(u) − p(v) between the points p(u) and p(v).

Bill Jackson Rigidity of Graphs and Frameworks

Rigidity and Global Rigidity

Two d-dimensional frameworks (G, p) and (G, q) are: equivalent if p(u) − p(v) = q(u) − q(v) for all uv ∈ E; congruent if p(u) − p(v) = q(u) − q(v) for all u, v ∈ V .

Bill Jackson Rigidity of Graphs and Frameworks

Rigidity and Global Rigidity

Two d-dimensional frameworks (G, p) and (G, q) are: equivalent if p(u) − p(v) = q(u) − q(v) for all uv ∈ E; congruent if p(u) − p(v) = q(u) − q(v) for all u, v ∈ V . A framework (G, p) is: globally rigid if every framework which is equivalent to (G, p) is congruent to (G, p);

Bill Jackson Rigidity of Graphs and Frameworks

Rigidity and Global Rigidity

Two d-dimensional frameworks (G, p) and (G, q) are: equivalent if p(u) − p(v) = q(u) − q(v) for all uv ∈ E; congruent if p(u) − p(v) = q(u) − q(v) for all u, v ∈ V . A framework (G, p) is: globally rigid if every framework which is equivalent to (G, p) is congruent to (G, p); rigid if there exists an ǫ > 0 such that every framework (G, q) which is equivalent to (G, p) and satisfies p(v) − q(v) < ǫ for all v ∈ V , is congruent to (G, p). (This is equivalent to saying that every continuous motion of the vertices of (G, p) in Rd, which preserves the lengths of all edges of (G, p), also preserves the distances between all pairs of vertices of (G, p).)

Bill Jackson Rigidity of Graphs and Frameworks

Rigidity: Example

t t t t t t t t

v1 v2 v3 v4 v1 v4 v2 v3 (G, p0) (G, p1)

Figure: The framework (G, p1) can be obtained from (G, p0) by a continuous motion in R2 which preserves all edge lengths, but changes the distance between v1 and v3. Thus (G, p0) is not rigid.

Bill Jackson Rigidity of Graphs and Frameworks

Global Rigidity: Example

✘ ✘ ✘ ✘ ✘ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

v1 v2 v4 e4 e1 e5 v1 v2 v4 e4 e1 e5 v3 v3 e2 e2 e3 e3

Figure: A rigid 2-dimensional framework which is not globally rigid. Corresponding edges in both frameworks have the same length, but the distances from v1 to v3 are different.

Bill Jackson Rigidity of Graphs and Frameworks

Complexity

It is NP-hard to determine whether a given d-dimensional framework (G, p) is globally rigid for d ≥ 1 (J. B. Saxe), or rigid for d ≥ 2 (Abbot).

Bill Jackson Rigidity of Graphs and Frameworks

Complexity

It is NP-hard to determine whether a given d-dimensional framework (G, p) is globally rigid for d ≥ 1 (J. B. Saxe), or rigid for d ≥ 2 (Abbot). These problems becomes more tractable if we restrict attention to ‘generic’ frameworks (those for which the set of coordinates of all points p(v), v ∈ V , is algebraically independent over Q).

Bill Jackson Rigidity of Graphs and Frameworks

The Rigidity Matrix

The rigidity matrix R(G, p) of a d-dimensional framework (G, p) is the |E| × d|V | matrix with rows indexed by E and sequences of d consecutive columns indexed by V .

Bill Jackson Rigidity of Graphs and Frameworks

The Rigidity Matrix

The rigidity matrix R(G, p) of a d-dimensional framework (G, p) is the |E| × d|V | matrix with rows indexed by E and sequences of d consecutive columns indexed by V . The entries in the row corresponding to an edge e ∈ E and columns corresponding to a vertex u ∈ V are given by the vector p(u) − p(v) if e = uv is incident to u and is the zero vector if e is not incident to u.

Bill Jackson Rigidity of Graphs and Frameworks

Rigidity matrix: Example

r r r r

v1 v2 v3 v4 e1 e2 e3 e4    

v1 v2 v3 v4 e1

p(v1) − p(v2) p(v2) − p(v1)

e2

p(v2) − p(v3) p(v3) − p(v2)

e3

p(v3) − p(v4) p(v4) − p(v3)

e4

p(v1) − p(v4) p(v4) − p(v1)    

Bill Jackson Rigidity of Graphs and Frameworks

Infinitesimal Motions

Each vector q in the null space of R(G, p) is an infinitesimal motion of (G, p). Taking q : V → Rd we have [q(u) − q(v)] · [p(u) − p(v)] for all e = uv ∈ E so the vectors q(u) are instantaneous velocities which preserve lengths of edges.

Bill Jackson Rigidity of Graphs and Frameworks

Infinitesimal Motions

Each vector q in the null space of R(G, p) is an infinitesimal motion of (G, p). Taking q : V → Rd we have [q(u) − q(v)] · [p(u) − p(v)] for all e = uv ∈ E so the vectors q(u) are instantaneous velocities which preserve lengths of edges. Since each continuous isometry of Rd gives rise to an infinitesimal motion of (G, p), the dimension of the kernal of R(G, p) is at least d+1

2

• whenever p(V ) affinely spans Rd.

Bill Jackson Rigidity of Graphs and Frameworks

Infinitesimal Motions

Each vector q in the null space of R(G, p) is an infinitesimal motion of (G, p). Taking q : V → Rd we have [q(u) − q(v)] · [p(u) − p(v)] for all e = uv ∈ E so the vectors q(u) are instantaneous velocities which preserve lengths of edges. Since each continuous isometry of Rd gives rise to an infinitesimal motion of (G, p), the dimension of the kernal of R(G, p) is at least d+1

2

• whenever p(V ) affinely spans Rd.

Hence rank R(G, p) ≤ d|V | − d + 1 2

• ,

and (G, p) will be rigid if equality holds.

Bill Jackson Rigidity of Graphs and Frameworks

Infinitesimal Motions

Each vector q in the null space of R(G, p) is an infinitesimal motion of (G, p). Taking q : V → Rd we have [q(u) − q(v)] · [p(u) − p(v)] for all e = uv ∈ E so the vectors q(u) are instantaneous velocities which preserve lengths of edges. Since each continuous isometry of Rd gives rise to an infinitesimal motion of (G, p), the dimension of the kernal of R(G, p) is at least d+1

2

• whenever p(V ) affinely spans Rd.

Hence rank R(G, p) ≤ d|V | − d + 1 2

• ,

and (G, p) will be rigid if equality holds. We say that G is infinitesimally rigid if rank R(G, p) = d|V | − d + 1 2

• .

Bill Jackson Rigidity of Graphs and Frameworks

Generic Rigidity and Independence

Theorem [Asimow and Roth, 1979] Suppose (G, p) is a generic d-dimensional framework with n ≥ d + 1 vertices. Then (G, p) is rigid if and only if it is infinitesimally rigid.

Bill Jackson Rigidity of Graphs and Frameworks

Generic Rigidity and Independence

Theorem [Asimow and Roth, 1979] Suppose (G, p) is a generic d-dimensional framework with n ≥ d + 1 vertices. Then (G, p) is rigid if and only if it is infinitesimally rigid. This implies that the rigidity of a generic framework (G, p) is determined by the rank of its rigidity matrix and hence depends

• nly on the graph G.

Bill Jackson Rigidity of Graphs and Frameworks

Generic Rigidity and Independence

Theorem [Asimow and Roth, 1979] Suppose (G, p) is a generic d-dimensional framework with n ≥ d + 1 vertices. Then (G, p) is rigid if and only if it is infinitesimally rigid. This implies that the rigidity of a generic framework (G, p) is determined by the rank of its rigidity matrix and hence depends

• nly on the graph G.

We say that G = (V , E) is independent in Rd if the rows of R(G, p) are linearly independent for any generic (G, p). Similarly F ⊆ E is independent if the rows of R(G, p) indexed by F are linearly independent

Bill Jackson Rigidity of Graphs and Frameworks

Generic Rigidity and Independence

Theorem [Asimow and Roth, 1979] Suppose (G, p) is a generic d-dimensional framework with n ≥ d + 1 vertices. Then (G, p) is rigid if and only if it is infinitesimally rigid. This implies that the rigidity of a generic framework (G, p) is determined by the rank of its rigidity matrix and hence depends

• nly on the graph G.

We say that G = (V , E) is independent in Rd if the rows of R(G, p) are linearly independent for any generic (G, p). Similarly F ⊆ E is independent if the rows of R(G, p) indexed by F are linearly independent If we can determine (generic) independence in Rd then we can determine (generic) rigidity in Rd.

Bill Jackson Rigidity of Graphs and Frameworks

Maxwell’s condition and Laman’s Theorem

Given a graph G and X ⊆ V let i(X) denote the number of edges

• f G joining the vertices of X.

Lemma [Maxwell’s Condition] If G is independent in Rd then i(X) ≤ d|X| − d+1

2

• for all X ⊆ V

with |X| ≥ d + 1.

Bill Jackson Rigidity of Graphs and Frameworks

Maxwell’s condition and Laman’s Theorem

Given a graph G and X ⊆ V let i(X) denote the number of edges

• f G joining the vertices of X.

Lemma [Maxwell’s Condition] If G is independent in Rd then i(X) ≤ d|X| − d+1

2

• for all X ⊆ V

with |X| ≥ d + 1. It is straightforward to show this condition characterises independence when d = 1. Laman showed this also holds when d = 2. Theorem [Laman, 1970] G is independent in R2 if and only if i(X) ≤ 2|X| − 3 for all X ⊆ V with |X| ≥ 2.

Bill Jackson Rigidity of Graphs and Frameworks

The Lov´ asz-Yemini Theorem

Laman’s Theorem and a classical result of Edmunds from matroid theory characterise generic rigidity when d = 2. We need the following concept. A t-thin cover of a graph G is a family X of subsets of V of size at least two such that each edge of G is induced by at least one set in X and |Xi ∩ Xj| ≤ t for all distinct Xi, Xj ∈ X.

Bill Jackson Rigidity of Graphs and Frameworks

The Lov´ asz-Yemini Theorem

Laman’s Theorem and a classical result of Edmunds from matroid theory characterise generic rigidity when d = 2. We need the following concept. A t-thin cover of a graph G is a family X of subsets of V of size at least two such that each edge of G is induced by at least one set in X and |Xi ∩ Xj| ≤ t for all distinct Xi, Xj ∈ X. Theorem [Lov´ asz and Yemini, 1982] G is rigid in R2 if and only if

• X∈X

(2|X| − 3) ≥ 2|V | − 3 for all 1-thin covers X of G.

Bill Jackson Rigidity of Graphs and Frameworks

Example

r r r r r r r r

v1 v5 v2 v6 v3 v7 v4 v8 Let X = {X1, X2, X3, X4} where X1 = {v1, v2, v3, v4}, X2 = {v5, v6, v7, v8}, X3 = {v2, v5} and X4 = {v3, v8}. Then

• X∈X

(2|X| − 3) = 5 + 5 + 1 + 1 = 12 < 2|V | − 3 so G is not rigid in R2.

Bill Jackson Rigidity of Graphs and Frameworks

Independence in R3

The banana graph shows that Maxwell’s condition, i(X) ≤ 3|X| − 6 for all X ⊆ V with |X| ≥ 4, does not characterise generic independence in R3.

❍❍ ✟✟✡ ✡ ✡ ✟ ✟ ✟ ✟ ❅ ❅ ❅ ❅ ❍ ❍ ❍ ❍

• ❍❍❍❍

✟ ✟ ❍❍

• ✡

✡ ✡ ❏ ❏ ❏ ❅ ❅ ❅ ❅ ✟ ✟ ✟ ✟ ❏ ❏ ❏ s s s s s s s s

v1 v2 v3 v4 v5 v6 v7 v8 |E| = 18 = 3|V | − 6

Bill Jackson Rigidity of Graphs and Frameworks

Independence in R3

❍❍ ✟✟✡ ✡ ✡ ✟ ✟ ✟ ✟ ❅ ❅ ❅ ❅ ❍ ❍ ❍ ❍

• ❍❍❍❍

✟ ✟ ❍❍

• ✡

✡ ✡ ❏ ❏ ❏ ❅ ❅ ❅ ❅ ✟ ✟ ✟ ✟ ❏ ❏ ❏ s s s s s s s s

v1 v2 v3 v4 v5 v6 v7 v8 |E| = 18 = 3|V | − 6 We can use 2-thin covers to show that G is not independent in R3. Let e = v1v2, X1 = {v1, v2, v3, v4, v5} and X2 = {v1, v2, v6, v7, v8}. Choose a maximum independent set I of edges of G + e which contains e. Let Ii = I ∩ E(Xi). Then |Ii| ≤ 3|Xi| − 6 = 9 for i = 1, 2. Since e ∈ I1 ∩ I2 we have I ≤ 17. Hence rank R(G + e, p) ≤ 17 for all (generic) p.

Bill Jackson Rigidity of Graphs and Frameworks

The Dress Conjecture

Dress conjectured that 2-thin covers can be used to characterise generic rigidity in R3. A hinge of a 2-thin cover X is a pair of sets Xi, Xj ∈ X such that |Xi ∩ Xj| = 2. Let h(X) denote the number

• f hinges of X.

Bill Jackson Rigidity of Graphs and Frameworks

The Dress Conjecture

Dress conjectured that 2-thin covers can be used to characterise generic rigidity in R3. A hinge of a 2-thin cover X is a pair of sets Xi, Xj ∈ X such that |Xi ∩ Xj| = 2. Let h(X) denote the number

• f hinges of X.

Conjecture [Dress, 1983] G = (V , E) is rigid in R3 if and only if

• X∈X

f (X) − h(X) ≥ 3|V | − 6 (1) for all 2-thin covers X of G, where f (X) = 1 if |X| = 2 and

• therwise f (X) = 3|X| − 6.

Bill Jackson Rigidity of Graphs and Frameworks

The Dress Conjecture

Dress conjectured that 2-thin covers can be used to characterise generic rigidity in R3. A hinge of a 2-thin cover X is a pair of sets Xi, Xj ∈ X such that |Xi ∩ Xj| = 2. Let h(X) denote the number

• f hinges of X.

Conjecture [Dress, 1983] G = (V , E) is rigid in R3 if and only if

• X∈X

f (X) − h(X) ≥ 3|V | − 6 (1) for all 2-thin covers X of G, where f (X) = 1 if |X| = 2 and

• therwise f (X) = 3|X| − 6.

Unfortunately this conjecture is FALSE - together with Jord´ an, we showed in 2005 that the LHS of (1) is negative when |V | is large. (The problem is that 2-thin covers can have lots of hinges.)

Bill Jackson Rigidity of Graphs and Frameworks

Modified conjecture

The conjecture may become true if we restrict the number of hinges in the cover. The hinge graph of a 2-thin cover X is the graph HX with vertex set X, in which two sets Xi, Xj ∈ X are adjacent if they intersect in a hinge. The cover X is k-degenerate if HX can be reduced to the empty graph by recursively deleting vertices of degree at most k.

Bill Jackson Rigidity of Graphs and Frameworks

Modified conjecture

The conjecture may become true if we restrict the number of hinges in the cover. The hinge graph of a 2-thin cover X is the graph HX with vertex set X, in which two sets Xi, Xj ∈ X are adjacent if they intersect in a hinge. The cover X is k-degenerate if HX can be reduced to the empty graph by recursively deleting vertices of degree at most k. Conjecture G = (V , E) is rigid in R3 if and only if

• X∈X

f (X) − h(X) ≥ 3|V | − 6 (2) for all 9-degenerate, 2-thin covers X of G.

Bill Jackson Rigidity of Graphs and Frameworks

Modified conjecture

The conjecture may become true if we restrict the number of hinges in the cover. The hinge graph of a 2-thin cover X is the graph HX with vertex set X, in which two sets Xi, Xj ∈ X are adjacent if they intersect in a hinge. The cover X is k-degenerate if HX can be reduced to the empty graph by recursively deleting vertices of degree at most k. Conjecture G = (V , E) is rigid in R3 if and only if

• X∈X

f (X) − h(X) ≥ 3|V | − 6 (2) for all 9-degenerate, 2-thin covers X of G. We can at least show that necessity holds i.e. inequality (2) must hold if G is rigid.

Bill Jackson Rigidity of Graphs and Frameworks

Example

r r r r r r r r r r r r r r r r r r r r r ❅ ❅ ☞ ☞ ☞ ☞

Let X = {X1, X2, . . . , X7} where G[Xi] = K5. Then HX = C7 so X is a 2-degenerate, 2-thin cover. We have

• X∈X

f (X) − h(X) = 7 × 9 − 7 = 56 < 3|V | − 6 so G is not rigid in R3.

Bill Jackson Rigidity of Graphs and Frameworks

Global Rigidity - Stress Matrix

A stress for a framework (G, p) is a map w : E → R such that, for all v ∈ V ,

• uv∈E

we(p(u) − p(v)) = 0.

Bill Jackson Rigidity of Graphs and Frameworks

Global Rigidity - Stress Matrix

A stress for a framework (G, p) is a map w : E → R such that, for all v ∈ V ,

• uv∈E

we(p(u) − p(v)) = 0. The associated stress matrix S(G, p, w) is the n × n matrix with rows and columns indexed by V in which the entry corresponding to an edge uv ∈ E is −we, all other off-diagonal entries are zero, and the diagonal entries are chosen to give zero row and column sums.

Bill Jackson Rigidity of Graphs and Frameworks

Stress Matrix: Example

s s s s

1 1 1 1

• 1
• 1

v1 v2 v3 v4    

v1 v2 v3 v4 v1

1 −1 1 −1

v2

−1 1 −1 1

v3

1 −1 1 −1

v4

−1 1 −1 1    

Bill Jackson Rigidity of Graphs and Frameworks

Generic Global Rigidity: Stress Matrix Characterisation

Theorem [Connelly (2005); Gortler, Healy and Thurston (2010)] Let (G, p) be a d-dimensional framework with n ≥ d + 2 vertices. Then rank S(G, p, w) ≤ n − d − 1 for all stresses w for (G, p). When (G, p) is generic, (G, p) is globally rigid if and only if (G, p) has a stress w such that rank S(G, p, w) = n − d − 1.

Bill Jackson Rigidity of Graphs and Frameworks

Generic Global Rigidity: Stress Matrix Characterisation

Theorem [Connelly (2005); Gortler, Healy and Thurston (2010)] Let (G, p) be a d-dimensional framework with n ≥ d + 2 vertices. Then rank S(G, p, w) ≤ n − d − 1 for all stresses w for (G, p). When (G, p) is generic, (G, p) is globally rigid if and only if (G, p) has a stress w such that rank S(G, p, w) = n − d − 1. This result implies that the global rigidity of a generic framework (G, p) depends only on the graph G. We say that G is globally rigid in Rd if some (or equivalently, every) generic d-dimensional framework (G, p) is globally rigid.

Bill Jackson Rigidity of Graphs and Frameworks

Generic Global rigidity: Combinatorial Conditions

Theorem [Hendrickson (1992)] Suppose G is globally rigid in Rd. Then either G is a complete graph on at most d + 1 vertices, or G is (d + 1)-connected and redundantly rigid i.e. G − e is rigid in Rd for all e ∈ E.

Bill Jackson Rigidity of Graphs and Frameworks

Generic Global rigidity: Combinatorial Conditions

Theorem [Hendrickson (1992)] Suppose G is globally rigid in Rd. Then either G is a complete graph on at most d + 1 vertices, or G is (d + 1)-connected and redundantly rigid i.e. G − e is rigid in Rd for all e ∈ E. These necessary conditions for global rigidity are sufficient when d = 1 and when d = 2 (Connelly, 2005; Jackson and Jord´ an, 2005). They are not sufficient for d ≥ 3. However, Tanigawa has shown that a slightly stronger condition is sufficient for all d.

Bill Jackson Rigidity of Graphs and Frameworks

Generic Global rigidity: Combinatorial Conditions

Theorem [Hendrickson (1992)] Suppose G is globally rigid in Rd. Then either G is a complete graph on at most d + 1 vertices, or G is (d + 1)-connected and redundantly rigid i.e. G − e is rigid in Rd for all e ∈ E. These necessary conditions for global rigidity are sufficient when d = 1 and when d = 2 (Connelly, 2005; Jackson and Jord´ an, 2005). They are not sufficient for d ≥ 3. However, Tanigawa has shown that a slightly stronger condition is sufficient for all d. Theorem [Tanigawa (2015)] G is globally rigid in Rd if G − v is rigid in Rd for all v ∈ V .

Bill Jackson Rigidity of Graphs and Frameworks

Body-Bar Frameworks

A d-dimensional body-bar framework is a collection of d-dimensional rigid bodies in Rd linked by disjoint bars which fix the distances between their end-points. The underlying graph of this framework represents bodies by vertices and bars by edges.

Bill Jackson Rigidity of Graphs and Frameworks

Body-Bar Frameworks

A d-dimensional body-bar framework is a collection of d-dimensional rigid bodies in Rd linked by disjoint bars which fix the distances between their end-points. The underlying graph of this framework represents bodies by vertices and bars by edges. Theorem [Tay (1991); Connelly, Jord´ an and Whiteley (2013)] A generic d-dimensional body-bar framework is rigid if and only if its underlying graph G has d+1

2

• edge-disjoint spanning trees; it is

globally rigid if and only if it is redundantly rigid.

Bill Jackson Rigidity of Graphs and Frameworks

Direction-Length Frameworks

A d-dimensional direction-length framework is a collection of points in Rd linked by constraints which fix either the direction or distance between some pairs of points. The underlying graph of this framework is a ‘mixed’ graph G = (V , ED ∪ EL).

Bill Jackson Rigidity of Graphs and Frameworks

Direction-Length Frameworks

A d-dimensional direction-length framework is a collection of points in Rd linked by constraints which fix either the direction or distance between some pairs of points. The underlying graph of this framework is a ‘mixed’ graph G = (V , ED ∪ EL). Theorem [Servatius and Whiteley (1999)] A mixed graph G is independent in R2 if and only if, for all ∅ = X ⊆ V , we have i(X) ≤ 2|X| − 2, iD(X) ≤ 2|X| − 3, and iL(X) ≤ 2|X| − 3.

Bill Jackson Rigidity of Graphs and Frameworks

Direction-Length Frameworks

A d-dimensional direction-length framework is a collection of points in Rd linked by constraints which fix either the direction or distance between some pairs of points. The underlying graph of this framework is a ‘mixed’ graph G = (V , ED ∪ EL). Theorem [Servatius and Whiteley (1999)] A mixed graph G is independent in R2 if and only if, for all ∅ = X ⊆ V , we have i(X) ≤ 2|X| − 2, iD(X) ≤ 2|X| − 3, and iL(X) ≤ 2|X| − 3. Together with Clinch and Keevash, we have recently characterised mixed graphs with the property that all their 2-dimensional generic realisations are globally rigid.

Bill Jackson Rigidity of Graphs and Frameworks

Point-Line Frameworks

A point-line framework is a collection of points and lines in R2 linked by constraints which fix the distances between some pairs of points and some pairs of points and lines, and the angles between some pairs of lines. Let G = (VP ∪ VL, E) be the underlying graph

• f this framework. For A ⊂ E, let νP, νL denote the numbers of

point and line vertices incident with A.

Bill Jackson Rigidity of Graphs and Frameworks

Point-Line Frameworks

A point-line framework is a collection of points and lines in R2 linked by constraints which fix the distances between some pairs of points and some pairs of points and lines, and the angles between some pairs of lines. Let G = (VP ∪ VL, E) be the underlying graph

• f this framework. For A ⊂ E, let νP, νL denote the numbers of

point and line vertices incident with A. Theorem[Jackson and Owen (2016)] A point-line graph G is independent if and only if, for all ∅ = A ⊆ E and all partitions {A1, . . . , As} of A, |A| ≤

s

• i=1

(2νP(Ai) + νL(Ai) − 2) + νL(A) − 1.

Bill Jackson Rigidity of Graphs and Frameworks

Matrix Completability

In this setting, two d-dimensional frameworks (G, p) and (G, q) are: equivalent if p(u) · p(v) = q(u) · q(v) for all uv ∈ E; congruent if p(u) · p(v) = q(u) · q(v) for all u, v ∈ V . Local and global completability of (G, p) in Rd are defined in an analogous way to (local) and global rigidity using the new definitions of equivalence and congruence.

Bill Jackson Rigidity of Graphs and Frameworks

Matrix Completability

In this setting, two d-dimensional frameworks (G, p) and (G, q) are: equivalent if p(u) · p(v) = q(u) · q(v) for all uv ∈ E; congruent if p(u) · p(v) = q(u) · q(v) for all u, v ∈ V . Local and global completability of (G, p) in Rd are defined in an analogous way to (local) and global rigidity using the new definitions of equivalence and congruence. This terminology comes from the fact that (G, p) is globally completable if and only if the Gram matrix for the points {p(v) : v ∈ V } is uniquely defined by the entries corresponding to the edges of G.

Bill Jackson Rigidity of Graphs and Frameworks

Matrix Completability

Two necessary conditions for the generic independence of G in Rd are that: i(X) ≤ d|X| − d

2

• for all X ⊆ V with |X| ≥ d;

|E(H)| ≤ d|V (H)| − d2 for all bipartite subgraphs H ⊆ G with at least d vertices on each side of their bipartition.

Bill Jackson Rigidity of Graphs and Frameworks

Matrix Completability

Two necessary conditions for the generic independence of G in Rd are that: i(X) ≤ d|X| − d

2

• for all X ⊆ V with |X| ≥ d;

|E(H)| ≤ d|V (H)| − d2 for all bipartite subgraphs H ⊆ G with at least d vertices on each side of their bipartition. Singer and Cucirangu (2010) show that these conditions are sufficient to characterise generic independence (and hence local completability) when d = 1. They also show that a generic 1-dimensional framework (G, p) is globally completable if and only if G is connected and not bipartite.

Bill Jackson Rigidity of Graphs and Frameworks

Matrix Completability

Two necessary conditions for the generic independence of G in Rd are that: i(X) ≤ d|X| − d

2

• for all X ⊆ V with |X| ≥ d;

|E(H)| ≤ d|V (H)| − d2 for all bipartite subgraphs H ⊆ G with at least d vertices on each side of their bipartition. Singer and Cucirangu (2010) show that these conditions are sufficient to characterise generic independence (and hence local completability) when d = 1. They also show that a generic 1-dimensional framework (G, p) is globally completable if and only if G is connected and not bipartite. The above necessary conditions for generic independence are not sufficient when d ≥ 2. It seems to be a difficult problem to characterise generic local or global completability even when d = 2. (Together with Jord´ an and Tanigawa (2016), we show that global completability is NOT a generic property when d = 2.)

Bill Jackson Rigidity of Graphs and Frameworks